3.1 introduction - heriot-watt university
TRANSCRIPT
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3 Chapter 3
DESCRIPTION OF THE PORE NETWORK SIMULATOR
3.1 Introduction
Strong evidence has emerged from a series of numerical and experimental studies that
the conceptual framework used in traditional modelling approaches oversimplifies the
physics of a number of important multiphase flow phenomena in porous media. The
generalised Darcy flow continuum assumption implemented in most standard simulators
has been shown to be poorly suited for adequately characterising gravity-destabilised
unsteady-state migratory flow or the dendritic growth pattern of viscous fingers under
adverse viscosity ratios. Pore network modelling (PNM) is an alternative modelling
approach that avoids these difficulties by permitting a less constrained examination of
the interactions between the pore structure, fluid properties and flow conditions. It is an
in silico approach that employs a localised representation of the physics of fluid flow in
porous media on a lattice of interconnected porous elements with carefully assigned
properties such as lengths, diameters, etc. This chapter describes a pore network model
that incorporates the pertinent physics governing multiphase flow in porous media under
the influence of capillary, gravity and viscous forces in the context of external drive
(injection) and internal drive (solution gas drive) processes. It is an extension of the quasi-
static model originally developed by McDougall and co-workers (McDougall and Sorbie,
1999; McDougall and Mackay, 1998; Bondino et al., 2005b; Ezeuko, 2009) to study
depressurization and repressurization processes. The extension of the model to the
analysis of external drive processes such as CO2 injection for Storage and EOR motivated
the development of new algorithms for advancing fluid interfaces during steady-state
flow that includes a more rigorous coupling of viscous effects. Modifications have also
been applied to the suite of algorithms associated with buoyancy-driven unsteady-state
migration to facilitate analysis of more complex flow processes such as matrix to fracture
flow during evolution of solution gas. The enhanced model features increases both the
Chapter 3 - Description of the Pore Network Simulator
84
range of multiphase flow phenomena that can be interpreted and the accuracy of such
interpretations.
The chapter is divided into roughly two parts, each describing the flow equations and the
outlines of implementation algorithms of subsets of closely related process features.
The first part describes the suite of modules dealing with the implementation of pore
level mechanisms observed during experiments of solution gas drive: bubble nucleation,
diffusive mass transport, bubble growth, coalescence, oil shrinkage, etc. The second part
deals with modules that are inherent to but not exclusive to external drive processes.
They include: a dynamic model of steady-state drainage with coupled capillary, gravity
and viscous forces; spontaneous cluster migration under buoyancy and viscous forces; a
model of gas dissolution; implementation of an EOS-based model of CO2 compressibility
factor and CO2 density, and the implementation of a model of CO2 solubility in brine by
Duan et al (2006). Note that these groupings have been done for descriptive purposes
only as any relevant module in the code can be accessed by both the internal and external
drive processes depending on the simulation settings and the dictates of the process
physics β a depressurization simulation in which gravity force has been enabled will, for
example, routinely call the spontaneous migration module.
First, a brief description of the generic pore space representation scheme used by the
model is given, along with a review of the phase clustering algorithm.
3.2 Pore Space Representation, Clustering and Trapping
Figure 3-1 shows a 3D lattice representation of a network model. The bonds are assumed
to both hold and transmit the fluids i.e. no distinction is made between pores and throats.
The network is defined by a few basic parameters:
β’ Its size β defined by Nx x Ny x Nz where Nx, Ny, Nz are the number of nodes in the
x, y and z directions respectively. For a 2D network Nz = 1;
β’ Its pore size distribution β bonds are randomly assigned radii from a well-defined
statistical distribution function;
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85
Its average coordination number, Z β this is the average number of bonds
connected to a node and represents the level of connectivity. In the networks
used here the maximum Z for a 3D network is 6, while a 2D network has a
maximum Z of 4. Physical porous media tend to exhibit a range of connectivity and
more realistic representations can be considered by randomly removing a fraction
of the bonds from the fully-connected template. Alternatively, pore size
distribution and connectivity data extracted from physical porous media via
mercury intrusion experiments or void space image reconstruction techniques can
be used to build a topologically equivalent network.
Figure 3-1: A three-dimensional network model. Pore radii are distributed according to a given distribution function.
3.2.1 Clustering and Trapping
Clustering is the procedure used during two-phase and three phase flow simulation to
keep track of contiguous groups of pores filled with a fluid phase of the same type that is
different from those of the surrounding pores. Phase clusters are assigned unique labels
that distinguish them as short-lived autonomous entities with boundaries that evolve as
fluid interfaces are created, destroyed or transferred β through drainage, imbibition,
coalescence, fragmentation events β in the course of a flow process. Clustering makes
possible the rule-based implementation of fundamental physics (e.g. multiphase
diffusion, bubble coalescence, etc) and the calculation of important petrophysical
quantities (critical gas saturation, relative permeability) on network models. Since every
displacement event has the potential to change the cluster landscape (cluster numbers,
size distributions, configurations), especially in highly volatile processes such as buoyancy-
driven migration, an efficient clustering algorithm is indispensable. The algorithm adapted
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86
here is an extension to 3D, of a 2D clustering algorithm first presented by Hoshen and
Kopelman (1976). The algorithm avoids repetitive re-scanning of the whole network
during the labelling process and thereby greatly reduces clustering CPU demand.
A cluster is in general considered trapped if none of its constituent pores is connected to
either the network inlet or outlet. Note that by allowing the production of the liquid
substrate (oil or water) at both ends of the network, the topological constraints on gas
bubble growth is relaxed, and (following experimental observations at Heriot-Watt) the
possible overestimation of average liquid saturations post-drainage can be avoided.
Having briefly outlined the generic network architecture and reviewed some
computational issues, we are ready to discuss the physics governing the different physical
processes and how they have been implemented in the model. Largely depressurization-
related processes will be discussed next (sections 3.3 to 3.7), followed by processes
primarily relevant to injection. Except under three-phase conditions, descriptions of
depressurization processes are given in terms of oil and gas phases, whilst injection
processes are presented mainly with respect to the multiphase flow of gas (or CO2) and
water.
3.3 Bubble Nucleation during Depressurization
Depressurization is a hydrocarbon recovery process that involves decreasing the reservoir
pressure (via fluid withdraw) in order to expel oil, firstly, by the expansion of the oil itself
and, secondly, by the displacement of the oil in place by liberated solution gas as
reservoir pressure drops below the bubble point. As pressure declines, the solubility of
gas in oil decreases. As pressure continues to decrease, there will eventually be more gas
in the oil than can be dissolved (i.e. it becomes supersaturated), and this excess gas must
be liberated from solution.
The appearance of embryonic bubbles marks the onset of a solution gas drive process.
Both instantaneous and progressive nucleation mechanisms have been implemented in
the model. In the case of instantaneous nucleation, a predefined number of nucleation
sites are activated simultaneously in a randomly chosen set of pores once the pressure
drops below the bubble point or after a specified average critical supersaturation is
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87
reached (difference between the actual dissolved gas concentration and that expected
from equilibrium PVT measurements).
The progressive nucleation mechanism implemented is based on the pre-existing vapour
model proposed by Yortos and Parlar (1989). In this model, each pore or a subset of the
pores is assumed to host a native crevice that acts as potential nucleation sites. Crevice
sizes are distributed according to a given distribution function, uncorrelated with the pore
sizes. A crevice of radius π is assumed to be activated when the local supersaturation
exceeds the capillary threshold of the crevice, viz:
(πΎπΆ β ππ) β₯2ππππ π
π (3-1)
where πΎ is the gas solubility constant, πΆ the local dissolved gas concentration, ππ
the local oil pressure, π is the gas/oil interfacial tension and π is the contact angle.
Hence, when a fluid system becomes supersaturated in a porous medium, the first pores
to host a bubble will be those containing the largest crevices. The critical supersaturation
for the onset of nucleation consequently depends upon the radius of the largest crevice,
whilst the overall degree of supersaturation during the process is a function of many
parameters: depletion rate, diffusivity, and the nucleation characteristics of the medium
(i.e. the crevice size distribution). Note that the term βcreviceβ density is simply used as a
physical cipher to describe sites more likely to host embryonic gas nuclei β i.e. sites of
increased nucleation potential (which could be due to impurities on the rock surface or
variations in mineralogy as opposed to pre-existing crevice bubbles).
Although there is some controversy as to which of the two nucleation mechanisms
dominate in the course of a depressurization process, it will be shown in Chapter 7 why
the instantaneous nucleation mechanism could be considered a special case of the
progressive mechanism occurring at a much faster rate. This is important because it is
often more efficient to use instantaneous nucleation mechanism to run parametric
sensitivity studies aimed at isolating specific interactions during depressurization.
Chapter 3 - Description of the Pore Network Simulator
88
3.4 Diffusive Mass Transport
The transfer of dissolved gas into a nucleated bubble is modelled through a first-
principles diffusion process. Upon nucleation, the dissolved gas concentration at the
gas/oil interface is assumed to instantaneously reach an equilibrium value that is
determined by the current system pressure. A dissolved gas concentration gradient is
thus set up throughout the entire system which drives dissolved gas towards the
embryonic bubble that acts as a gas sink. Now consider two adjacent pores π and π. The
mass of gas diffusing across unit cross-sectional area in unit time from pore π to pore π is
given by Fickβs first law:
π½ππ = βπ·(πΆπ β πΆπ)/πΏ (3-2)
where π½ππ is the mass flux from π to π, πΆπ the gas concentration in pore π, πΆπ the gas
concentration in pore π, D the diffusion coefficient, and L a diffusion length (taken
here to be equal to the distance between two pore centres).
Mass flux across an oil-filled pore will cause its concentration to change with time and this
time evolution of concentration is evaluated by discretizing Fickβs second law,
ππΆ
ππ‘=
π
ππ₯(π·
ππΆ
ππ₯) (3-3)
to the form
πΆππππ€ = πΆπ
πππ + βπ‘ [πππ π ππβπππ π ππ’π‘
ππ] =
βππ(βπ‘)
ππ (3-4)
where ππ is the pore volume and βπ‘ the time step.
For pore π in Figure 3-2, the area-weighted sum of diffusion fluxes βππ(βπ‘), is given as
βππ(βπ‘) = β π½π Γ π¦π’π§ (π΄π, π΄π)6π=1 (3-5)
Chapter 3 - Description of the Pore Network Simulator
89
where π΄ is the pore cross-sectional area and π an index running through all
perimeter pores.
The π¦π’π§ term in Equation (3-5) means that mass diffusion is constrained by the minimum
cross-sectional area between pores, making the peculiar characteristics of porous media
architecture (pore size distribution, coordination number) an important governing
parameter of a depletion process.
As gas is transferred into the bubble, local oil-filled gas concentrations will generally
decrease towards the equilibrium value at a rate that depends on the distance of the pore
from the bubble β pores closer to the bubble will approach equilibrium concentration
more rapidly than those farther away from it (Figure 3-3). Diffusion lengths, and thus the
time evolution of dissolved concentration distribution also depend on the overall network
connectivity.
The diffusion model just presented allows for both negative and positive mass fluxes into
a pore depending on the surrounding concentration gradients. This general feature makes
it fairly straightforward to adapt it for modelling both gas evolution during
depressurization, and gas dissolution during repressurization or during external gas
injection processes.
Figure 3-2: Schematic section of a pore network illustrating the diffusion model
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90
Figure 3-3: Time evolution of dissolved gas concentration contours due to mass transfer into five nucleated gas bubbles. Bubble growth was disabled for the purposes of illustration.
3.4.1 Diffusion in three-phase systems
The diffusion model is further generalized to handle mass diffusion in the presence of two
liquid phases with different diffusivities (e.g. oil and water). Diffusion between liquid
pores of the same phase is relatively straightforward but for diffusion across phase
boundaries (e.g. between oil and water), a more careful treatment is required. As well as
specifying self-diffusion coefficients (e.g. Doil-Doil, Dwater-Dwater), inter-phase diffusion
coefficients (e.g. Doil-Dwater, Dwater-Doil) must also be properly defined to ensure mass
conservation. Moreover, because gas is generally more soluble in oil than in water,
artificial concentration gradients are avoided during oil-water or water-oil diffusion by
multiplying the gas concentration in water by a partition coefficient defined as:
πΎπ ππ =πΊππ
πΊππ (3-6)
where GOR is the dissolved gas to oil ratio and GWR the dissolved gas to water
ratio.
3.5 Depletion Rate and Bubble Density
The first-principles approach to diffusive mass transfer enables the time evolution of
pore-level gas concentrations β due to concentration gradients created following bubble
nucleation β to be tracked explicitly. There is therefore a tendency during
depressurization for concentrations in oil-filled pores to approach the pressure-
dependent equilibrium GOR or GWR. If all other variables are considered constant, the
degree to which this is achieved will greatly depend on the depletion rate β the slower
Chapter 3 - Description of the Pore Network Simulator
91
the depletion rate the closer the dissolved concentration will be to the equilibrium GOR
or GWR.
During a pressure step, the diffusion process is discretized as follows. First, the time
associated with the current step is calculated according to the specified depletion rate.
Then the βtransferable gas massβ (TGM) β the mass of gas expected to come out of
solution if the system were now shut-in and allowed to equilibrate β is calculated by
comparing the current equilibrium solubilities (GOR and GWR) with the dissolved gas
concentration at the end of the previous pressure step. Diffusion can then proceed
according to Equation (3-5) for the specified time. Note that the faster the depletion rate,
the smaller the time available for diffusion for a given pressure drop and the smaller the
amount of gas transferred into the bubble. At high depletion rates, therefore, the actual
and equilibrium concentration profiles will diverge with pressure decline (Figure 3-4b)
and the elevated levels of local supersaturations may then be sufficient to trigger the
nucleation of new bubbles, increasing the nucleation density.
At very low depletion rates on the other hand, diffusion occurs over a longer time period
for a given pressure drop, sometimes long enough for the total TGM to go into the
bubble. If TGM becomes zero during a diffusion process, mass transfer across gas-oil
interfaces is stopped, and diffusion over the remaining time occurs exclusively within the
oleic phase, acting to equilibrate the rest of the system (Figure 3-4a). This means that
local supersaturations would remain low, making extra nucleation unlikely.
It is very important for any model of depressurization to be able to capture the
dependency of nucleation density and depletion rate because it is one of the key
interactions used to characterize a depletion process. The diffusion model described here
allows these two variables to be coupled together in a manner consistent with
experimental observations.
Chapter 3 - Description of the Pore Network Simulator
92
GOR
t
GOR
t
actual path div ergesdiffusion restores equilibrium
(a) (b)
Figure 3-4: Behaviour of GOR with time for; (a) diffusion-dominated system at low depletion rate, and (b) depletion-dominated system at a high depletion rate.
3.6 Bubble Growth and Expansion
The bubble growth process is modeled in two stages β in keeping with experimental
observations of the depletion process in micromodels. The first stage involves the growth
of the embryonic bubble within the host oil-filled pore as its pressure increases above the
ambient pressure due to gas diffusing into it. The bubble is assumed to start out as a
spherical point at the pore centre, grows to meet the pore wall, and then continues to
expand along the wall. Once the bubble has expanded to fill the entire pore, the second
stage of growth can then begin. For the bubble to expand beyond its parent pore, its
internal pressure (defined in terms of the number of moles contained within it current
volume, given its compressibility factor at the current system temperature) must be
greater than the minimum of the capillary entry thresholds of the oil-filled neighboring
pores that are not trapped as expressed by Equation (3-7).
ππ > π¦π’π§ (πππ) + ππ (3-7)
i.e.
ππ β [π¦π’π§ (πππ) + ππ] = π¦π’π§(βππ
π) > 0 (3-8)
where, ππ = the absolute pressure of the gas phase = π(π, π)π(π‘)π π/π(π‘), n =
current number of gas moles, R = universal gas constant, π = current pressure = ππ
= the absolute pressure in the oil evaluated at the upper boundary of the network,
π = current temperature, π = gas compressibility factor, and π(π‘) = current gas
volume, πππ = the static equilibrium capillary pressure in an available perimeter
Chapter 3 - Description of the Pore Network Simulator
93
pore = πππππ π/ππ, π = interfacial tension, π = gas-oil contact angle=0.0, ππ = pore
radius, π = shape factor which is 2 for cylindrical pore elements, π‘ the time, and π =
an index running through all the available pores containing oil at the perimeter of
the expanding gas.
If Equation (3-8) is not satisfied, the bubble is labeled as βconstrainedβ and further growth
is deferred so that the pressure of the bubble can continue to rise as new gas is added to
it via diffusion whilst the interface is held in a quasi-static state.
When the bubble becomes βunconstrainedβ, i.e. if the condition in Equation (3-8) becomes
satisfied, the bubble expands to fill the appropriate perimeter pore and a two-pore gas
cluster is formed. The distribution of gas inside a cluster is assumed to be homogeneous.
It is updated regularly by partitioning it among the constituent pores on a volume-
weighted basis. The filling of a perimeter pore therefore causes the cluster pressure to
drop and it immediately becomes βconstrainedβ once more. The combined effects of
system pressure decline and the increase in cluster pressure due to continued transfer of
dissolved gas via diffusion allow the cluster to continue to expand into perimeter oil-filled
pores.
If more than one bubble had nucleated, the evolution of each is treated independently
but the model also accounts for the possibility of coalescence of initially isolated evolving
clusters. When in the course of growth two gas-filled pores belonging to different parent
clusters become neighbours, then the next call of the clustering algorithm labels the two
parent clusters as one and the total mass content of the combined clusters is re-
partitioned among the pores of the new larger cluster on a volume-weighted basis. This
means that the clustering algorithm has to be called after every pore-filling event to check
for coalescence (or fragmentation) events, thus highlighting the need for a very efficient
clustering algorithm.
The perturbative effect of gravity on bubble growth can be included in Equation (3-8) to
give the pore invasion criteria around a cluster as
π¦π’π§(βπππ
+ βππβ)π > 0 (3-9)
Chapter 3 - Description of the Pore Network Simulator
94
where,
βππβ = ππ
βπ β βπππ
is the net local gravity component
where,
ππβπ = ππππ»π (see Figure 3-5), stands for the local hydrostatic head due to
the oil column. It increases with depth, i.e. bubbles at top of the network
will grow faster than those at the bottom.
βπππ = βππβππ (see Figure 3-5), stands for the local buoyancy pressure. It
mainly controls the trajectory of bubble growth.
where π»π is the vertical distance between oil pore π and the top of
the network, βππ the vertical distance between oil pore π and the
bottom of neighbouring gas cluster π, βπ = ππ β ππ, ππ the gas
density, ππ the oil density, and π the gravitational acceleration.
The invasion criteria in Equation (3-9) can be further generalized to include the effect of
an externally imposed pressure gradient as
π¦π’π§(βπππ
+ βππβ + βππ
π£ππ )π > 0 (3-10)
were, βπππ£ππ is the local viscous pressure gradient across an oil pore π at the
perimeter of gas a cluster π.
The bubble growth model just described is an example of the classic invasion percolation
β the gas/oil interface advances one pore at a time in discrete jumps equivalent to the
length of a pore. For depressurization in light oil under relatively small gravity and viscous
pressure gradients, the invasion percolation model will give accurate results.
Nevertheless, even with the addition of the viscous pressure term to the pore invasion
criteria (Equation 3-10), the pore filling procedure described by the classic invasion
percolation (IP) will fail to capture observed pore scale displacement behaviour when the
viscous pressure term becomes of the same or higher order of magnitude as the capillary
and gravity terms. A large pressure gradient imposed externally or generated internally
during a high rate depressurization in heavy oil could drive menisci into multiple pores
simultaneously. A growth model that captures these dynamic effects has been developed
Chapter 3 - Description of the Pore Network Simulator
95
and will be presented in the context of flow during gas injection β the same principles will
be seen to also apply to bubble growth during depressurization. Meanwhile, the 2-phase
classic IP model of bubble growth will now be extended to 3-phase systems.
Figure 3-5: Schematic section of pore network illustrating the elevation references for calculating the effective local gravity pressure acting on oil-filled pores βaβ and βbβ during the growth of a nucleated gas bubble following depressurization.
3.6.1 Bubble Growth under Waterflooded Conditions
So far weβve only considered bubble growth in a network saturated with a single liquid
substrate i.e. oil, and the consequent uniformity of IFT distribution throughout the system
(assuming negligible gravity and viscous forces) means that the pore filling sequence is
simply a function of the pore size. Under 3-phase conditions, however, pore entry
thresholds are more complicated as they are not only dependent on the competition
between the three possible IFTs β πππ, πππ€, and πππ€ β, but also on the configuration of
fluid interfaces within a pore (existence of films) and on the saturation history (depending
on the prevailing trapping patterns, the growth of a bubble might involve the
displacement of multiple fluid interfaces simultaneously). Before giving the expressions
for capillary thresholds under specific interface configurations, a model of film flow as a
function of spreading coefficient is first presented.
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96
3.6.1.1 Spreading coefficient and film flow
The spreading coefficient (πΆπ ) is a property of 3-phase flow used to characterize the
hierarchy of fluids in a pore. It is a function of the three system IFTs and expresses the
balance of free surface energy of the phases at equilibrium. It is calculated as
πΆπ = πππ€ β (πππ + πππ€) (3-11)
where, πππ€is the gas-water interfacial tension, πππthe gas-oil interfacial tension,
and πππ€ the oil-water interfacial tension.
If πΆπ β₯ 0 and the system is completely water-wet, then oil will spontaneously spread on
the water substrate which in turn spreads on the solid surface, whilst the gas phase
resides in the centre of the pore (Figure 3-6a). If πΆπ < 0 on the other hand, there will exist
a 3-phase contact line as shown in Figure 3-6b.
The presence of a system-wide network of interconnected films of a spreading oil (πΆπ β₯
0), offers the possibility of continuous oil production even at very low oil saturations.
Micromodel experiments (Idris, 1990; Danesh et al, 1991) have shown that oil production
via film flow can be quite efficient. Other studies (Γren et al, 1992; Kalaydjian and Tixier,
1991; and Kalaydjian, 1992) have reported generally lower displacement efficiencies for
non-spreading systems (πΆπ < 0) compared to spreading systems (πΆπ > 0).
The model distinguishes between spreading and non-spreading systems by using different
oil trapping criteria. For πΆπ < 0, an oil ganglion is trapped if it is surrounded by pores
filled with gas or water. A bubble nucleated in the oil ganglion can grow β and thus allow
the oil to escape, only via a double drainage pathway in which the gas displaces the oil
whilst the oil displaces the water. For πΆπ β₯ 0, oil can escape if there is a contiguous
pathway of pores filled with gas or oil that extends to either end of the network.
Chapter 3 - Description of the Pore Network Simulator
97
GAS
OIL
WATER WATER
OIL
GAS
(a) (b)
Figure 3-6: Schematic of phase hierarchies at the pore scale for (a) positive spreading coefficient, and (b) negative spreading coefficient.
3.6.1.2 Capillary Thresholds in Waterflooded Systems
We are now ready to formulate the expressions for the effective capillary thresholds
required to displace a range of interface configurations for both spreading and non-
spreading systems. Figure 3-7 shows simple pairs of competing potential displacement
events and the associated threshold criteria under spreading and non-spreading
scenarios. If π 1 β€ π 3 (which is perhaps the most probable scenario given the strong
water-wetting) then the oil-filled pore (pore 3) will be preferentially invaded in both the
spreading and non-spreading systems. Possible conditions under which the water-filled
pore will be preferentially invaded can be worked out viz: For a spreading oil π 1 π 2β >
(1 + πππ€/πππ), and for a non-spreading oil π 1 π 2β > πππ€/πππ . However, more concrete
experimental evidence is needed to fully assess the practical significance of such interface
configurations.
We next examine competing pairs of double drainage events for spreading and non-
spreading systems in Figure 3-8. If in Figure 3-8 π 1 = π 4, then the entry criteria predicts
pore 3 to preferentially fill for a spreading oil (since π 1 < π 3, and πππ€ > [πππ + πππ€]).
For the non-spreading oil, however, the preferred direction of growth is not so obvious.
Pore 3 is preferentially filled under non-spreading conditions if π 4 π 3β < (πππ€ β
πππ€)/πππ.
A more dramatic effect of oil spreading on displacement efficiency can be illustrated using
the interface configuration in Figure 3-9. Although the oil in pore 2 seems to be trapped
between gas-filled pores 1 and 3, the model can nevertheless allow it to be displaced
through pore 4 via the spreading films that line the pore walls. For non-spreading oil no
Chapter 3 - Description of the Pore Network Simulator
98
such pathway exits and the oil in pore 2 remains trapped (it could perhaps still be
produced by the application of a large external pressure gradient).
Figure 3-10 compares saturation distribution graphics at selected stages of evolution of
multiple bubbles in test simulations of depressurization in a waterflooded network with
spreading and non-spreading oil. The model successfully reproduces the experimentally
observed effects of oil spreading on displacement efficiency. With a positive spreading
coefficient, the evolving gas clusters exhibit compact growth structures (due to the more
efficient displacement mechanism) compared to the dendritic structures in the case of a
negative spreading coefficient.
(a)
(b)
Figure 3-7: Schematic of interface configurations in adjacent pore elements showing competing single drainage events and the associated capillary thresholds for; (a) a spreading and, (b) a non-spreading system.
Chapter 3 - Description of the Pore Network Simulator
99
(a)
(b)
Figure 3-8: Schematic of interface configurations in adjacent pore elements showing alternative double and single drainage events and the associated capillary thresholds for; (a) a spreading and, (b) a non-spreading system.
(a)
(b)
Figure 3-9: Schematic of interface configurations in adjacent pore elements showing the effect of phase trapping on gas saturation distribution for a spreading coefficient that is; (a) positive, and (b) negative.
Chapter 3 - Description of the Pore Network Simulator
100
T1 T2 T3 T4
a. Spreading oil
b. Non-Spreading oil
Figure 3-10: Time evolution of nucleated gas bubbles in a waterflooded network in which the oil is (a) Spreading, and (b) Non-spreading.
3.7 Oil Production and Gas Repartitioning During Bubble Growth
Consider bubble growth in a 2-phase (gas-oil) system. The invasion of an oil-filled pore
that is situated away from the network outlet by gas will lead to the production of the
same volume of oil at the outlet port of the network (assuming oil compressibility is
negligible). Since dissolved gas concentrations will generally increase away from the gas-
oil interface due to diffusion, the produced oil will likely be more supersaturated than oil
at the gas-oil interface which had just been displaced by gas. This means that for every
advance of the gas-oil interface there is a cascade of oil-oil displacement events that
extend all the way to the outlet, with the less supersaturated oil displacing the more
supersaturated oil.
Because the bubble growth process is localized, convective oil movement has not been
explicitly modelled and this means that appropriate corrections have to be applied in
order to keep the dissolved gas concentrations consistent with the physics of the process
being studied. This is done by comparing the dissolved gas concentrations of the oil-filled
pores at the outlet port with the concentration of the oil-filled pore to be invaded and the
difference used to repartition the dissolved gas mass in the oil-filled pores in the network
after every pore invasion. This repartitioning procedure has also been extended to
waterflooded systems.
Chapter 3 - Description of the Pore Network Simulator
101
Attention now turns to the discussion of multiphase flow mechanisms more closely
associated with (but not exclusive to) injection processes. The following sections give a
detailed description of a dynamic model of gas flow β under both stable and migratory
flow conditions β that significantly improves upon the concepts already presented in
relation to bubble growth during depressurization.
3.8 Gas Injection under the Coupled Effects of Capillary, Gravity and
Viscous forces The classic invasion percolation (CIP) model of drainage (displacement of a wetting phase
by a non-wetting phase) advances fluid-fluid interfaces one pore at a time regardless of
the injection rate under consideration. A pore is either filled with one fluid phase or
another and no partial pore filling is possible. In addition, CIP ignores the contribution of
fluid viscosity to the time dependence of pore filling.
This section presents a modified version of the invasion percolation model of drainage
that implements time-dependent, simultaneous multi-pore, fluid-fluid interface tracking
that accounts for the effects of fluid viscosity on displacement and flow.
Figure 3-11 shows a pore surrounded by four neighbours (perimeter pores) into which is
injected a fixed mole of gas at regular intervals. Neglecting all other forces for the
moment, the gas will expand into any perimeter water-filled pore (water is considered
the wetting phase here) that is not currently trapped when the following condition is
satisfied:
ππ β ππ€ > πππ (3-12)
where, ππ€ = the absolute pressure in the liquid (water) evaluated at the upper
boundary of the network, and other terms have their former meaning. Note that
Equation (3-12) is identical in form to Equation (3-7).
The net driving pressure for the displacement then becomes
Chapter 3 - Description of the Pore Network Simulator
102
βπππ =ππ β (ππ€ + ππ
π) (3-13)
More generally the local net pressure driving gas into available perimeter water pores
should include gravity and viscous forces and may be expressed as:
βπππππ‘ = βππ
π + βππβ + βππ
π£ππ (3-14)
where,
βππβ = The net local gravity component as described by equation (3-9)
βπππ£ππ = the local viscous pressure gradient (the determination of βππ
π£ππ is
discussed in the next section).
The local meniscus velocities are evaluated as
π£π = ππβπππππ‘/π΄π (3-15)
where, ππ = pore conductivity = πππ4/8πππππΏπ , ππππ = a volume weighted
viscosity, πΏπ = pore length, and π΄π = pore cross-sectional area.
The time it takes a gas-water meniscus advancing at a pore-level velocity, π£π, to
completely fill a pore of total length, πΏπ is then given by
π‘π =πΏπ
π£π(1 β ππ) (3-16)
where, ππ is the normalized position of the meniscus along the pore (note that
initially all perimeter pores are water-filled and ππ = 0).
After each injection step, molar masses and pressures are updated and gas clusters are
expanded through water-filled pores that are available for displacement. The gas-water
interface in all other perimeter pores with positive net driving pressures are advanced by
updating the meniscus positions following
πππππ€ = ππ
πππ +π¦π’π§(π‘π)
π‘π(1 β ππ
πππ) (3-17)
Chapter 3 - Description of the Pore Network Simulator
103
where π¦π’π§(π‘π) is the minimum of the sorted pore filling times.
If π¦π’π§(π‘π) is less than or equal to the length of the injection timestep, it means the
meniscus has reached the end of the pore with the minimum filling time and this pore is
immediately given a gas-pore label.
Note that if π¦π’π§(π‘π) is greater than the time available in the injection step, menisci inside
pores with positive net driving pressures are still advanced according to Equation (3-17)
but no pores will become completely filled with gas in the course of the injection step.
The calculation of gas cluster pressures after the injection step, however, accounts for the
change in gas volume due to the meniscus advance into neighbouring oil pores which are
now partially occupied by gas. This evens out the motion of the gas-water interface and
prevents artificial build-up of the gas pressure when, instead of water, a high viscosity
liquid is being displaced.
Figure 3-11: 2D Schematic of the onset of gas injection into a pore surrounded by four neighbours (perimeter pores). The default orientation of the viscous and gravity gradients with respect to the networks axes are as indicated.
3.8.1 Finding the Viscous Pressure Component
In the idealised lattice structure that represents the porous medium, each pore element
or bond connects uniquely to two nodes, i.e. nodes are the junctions where bonds meet,
Chapter 3 - Description of the Pore Network Simulator
104
Equation (3-1). To determine the local viscous pressure components in Equation (3-14),
we must solve a pressure field to determine the pressures at the nodes as follows.
First, a volume-weighted effective viscosity is calculated for each pore element using:
ππππ = πππ + (1 β π)ππ (3-18)
The conductivity of a pore element connecting nodes π and π is calculated assuming
Poiseuilleβs flow, viz:
πππ =ππππ
4
8ππππππ
πΏππ
(3-19)
and the volume flux through the pore element becomes:
πππ = πππβππππ£ππ (3-20)
Conservation of mass at node (π) over all connecting pores (π) is given as:
β ππππ = 0 (3-21)
which, in terms of conductivities and nodal pressures, becomes:
β πππβππππ£ππ = 0π (3-22)
Equation (3-22) is invoked at each node to derive a set of linear equations the solution to
which gives the nodal pressures. We use a Dirichlet-type boundary condition and
therefore nodal pressures at both ends of the network must first be found. The pressure
gradient across the entire network, βππ£ππ
ππππππ, during an injection process is approximated
as follows.
The procedure involves two main stages.
1. Given the current phase saturation distribution, estimate the current effective
multiphase transmissibility of the network as follows:
Chapter 3 - Description of the Pore Network Simulator
105
a. Impose an arbitrary global pressure gradient on the network, βπβ
b. Determine nodal pressures by solving the pressure field as in Equations
(3-18) to (3-22)
c. Assuming steady-state flow, determine the instantaneous cumulative flow
rate at the outlet pores, πβ = β ππ, where π is an index running through all
outlet pores. Note that no interface movements occur during this
calculation and hence the fluid saturation distribution is unchanged.
d. Calculate effective transmissibility for the current injection step, π
(ππππ)π
=πβ
βπβ (3-23)
2. Estimate the global viscous pressure gradient at the beginning of the current
injection step, π (Note an βinjection stepβ is used here in place of the conventional
βtime stepβ).
a. If π = 1, i.e. the first injection step of the simulation (no gas has been
injected yet) then
(βππ£ππ
ππππππ)π
= ππππ‘/(ππππ)π
(3-24)
where, ππππ‘ is the pre-set volumetric injection rate.
Each gas injection simulation is initialised with a constant volumetric gas
injection rate which is then converted into a molar injection rate and
discretised. For example, if ππππππ‘ is the total pore volume of the inlet ports
of the network, then in an injection step, οΏ½ΜοΏ½ moles of gas must be injected
and allowed to expand over a time βπ‘ (where οΏ½ΜοΏ½ = ππππππ‘π/ππ π,
βπ‘ = ππππππ‘/ππππ‘, π = current system pressure or pressure in the water
phase, π = universal gas constant, π = current system temperature, and π
= gas compressibility factor).
b. If π > 1 (some gas has already expanded), then
(βππ£ππ
ππππππ)π
= (π)πβ1/(ππππ)π
(3-24b)
Chapter 3 - Description of the Pore Network Simulator
106
where, (π)πβ1 = (βππ)πβ1
βπ‘ =β the actual gas expansion rate
over the previous injection step and (βππ)πβ1
the gas expansion in
the previous injection step.
Because of interfacial and viscous resistances in the network, the actual
rate of gas expansion is always less than the pre-set volumetric injection
rate (ππππ‘) in Equation (3-24). Equation (3-24b) accounts for this disparity.
Once βππ£ππ
ππππππ for the current injection step is known it is set as the inlet pressure and the
outlet pressure thus becomes zero. The pressure field is recalculated through Equations
(3-18) to (3-22) so that the contribution of the local viscous pressure component to the
motion of the gas-water interface can be incorporated into Equation (3-14).
3.8.1.1 Sign convention for the local viscous pressure component
It is critical to have a consistent and sensible convention for the orientation of the local
viscous pressure gradient across liquid-filled pores at the perimeter of an expanding gas
cluster. The convention adopted here is illustrated in Figure 3-12. The local viscous
pressure gradient across a perimeter liquid-filled pore π connecting nodes π’ and π is
calculated as
βππ’ππ = ππ’
π β πππ (3-25)
where ππ’π is the pressure at upstream node π’ of the gas pore π and ππ
π the pressure
at the downstream node π (Figure 3-12).
Thus, the model makes no a priori assumptions about the local flow direction of the
defending phase (water) under viscous forces, but rather determines this dynamically
with reference to the current position of the gas-water interface (Figure 3-12).
Chapter 3 - Description of the Pore Network Simulator
107
Figure 3-12: Schematic section of network demonstrating the convention used to calculate local viscous pressure gradient across perimeter liquid-filled pores.
3.8.2 General note on the discretization of the injection process
A higher degree of accuracy is achieved with finer injection steps. The frequency of
viscous pressure field updates over a given saturation change was also found to have a
large effect on the displacement pattern. Ideally, the pressure field should be updated
whenever there is a change in fluid saturation distribution, i.e. after every meniscus
advance, but this is obviously impractical computationally. According to the level of
precision required, a local process refinement has been implemented. It divides the
injection step into smaller time steps at the end of which updates of the viscous pressure
field could be scheduled.
3.9 Inlet Port for Full-Face Gas Injection In line with experimental observations and practice, a meaningful modelling of quasi-
steady-state gas-liquid relative permeability (Kr) during gas injection requires that there
be a network-spanning liquid cluster from the onset of the injection process. The use of a
single point injection strategy (which involves designating a dangling inlet pore as an
injection port, giving it a gas pore label, and then adding a fixed quantity of gas to it at
regular intervals depending on the set injection rate) ensures that an inlet-to-outlet
connectivity of the liquid phase is maintained even as the gas fills the network and
eventually breaks through. Liquid Kr is thus likely to be non-zero through most of the
saturation change during a single-point injection process (Figure 3-13a).
Chapter 3 - Description of the Pore Network Simulator
108
If, however, a full-face injection port (a configuration more commonly used in laboratory
experiments) were modelled in exactly the same way as a single-point by simply
designating all the dangling inlet pores as gas pores into which gas can be injected, then
the liquid phase will be non-network-spanning from the outset and its Kr will remain zero
throughout the injection process (Figure 3-13b). Moreover, because of the likely
variations in the conductivities of the inlet pores, it is probably not the case that all inlet
pores contribute equally to the total amount of gas entering the network during a full-
face injection. Thus a model of the inlet port that sensibly mimics the process of gas entry
during full-face injection has been implemented as described below.
Each dangling inlet liquid-filled pore is assumed to be at the perimeter of an imaginary
neighbor of the same size (Figure 3-14[a]). A fixed mass of gas is then put into these
imaginary pores at regular intervals (consistent with the set injection rate). The
deposition of gas into a fixed volume of an imaginary pore gradually raises the gas
pressure until it eventually overcomes the entry threshold of the neighboring βrealβ inlet
pore. The gas then expands into the pore following the steps outlined above β i.e.
meniscus in each of the βrealβ inlet pores are advanced simultaneously whilst taking
account of pore conductivity, viscous pressure gradient, etc (Figure 3-14[b]).
Note that if the meniscus has advanced into an inlet pore but has not yet filled it, the
fractional volume of the real pore occupied by gas is used in the calculation of the gas
pressure in the corresponding imaginary pore. Once an inlet pore is completely filled with
gas, its corresponding imaginary neighbor is plugged and subsequent gas injections are
made directly into the real inlet pore (Figure 3-14[c-f]). If a meniscus fails to reach the end
of an inlet pore after depositing an amount of gas (into its corresponding imaginary
neighbor) with an equilibrium PVT volume that is twice its pore volume, then further gas
injection into the associated imaginary pore is halted (i1/r1 in Figure 3-14[d-f]). The
meniscus will continue to advance into the real pore but at an increasingly slower rate.
This model of full-face injection not only allows for the preferential filling of inlet pores
but also permits βslowβ inlet pores to be appropriately by-passed by the incoming gas.
Chapter 3 - Description of the Pore Network Simulator
109
Hence, the connectivity of the defending phase to the inlet is not automatically broken
from the onset of injection (Figure 3-13c).
a. Single-point Injection
Liquid spanning; Krl>0, Krg=0
Liquid spanning; Krl>0, Krg=0
Liquid Spanning; Krl>0, Krg=0
Liquid spanning; Krl>0, Krg>0
b. Simple Full-face Injection
Liquid non-spanning; Krl=0, Krg=0
Liquid non-spanning; Krl=0, Krg=0
Liquid non-spanning; Krl=0, Krg=0
Liquid non-spanning; Krl=0, Krg>0
c. Dynamic Full-face Injection
Liquid spanning; Krl>0, Krg=0
Liquid spanning; Krl>0, Krg=0
Liquid spanning; Krl>0, Krg=0
Liquid Non-spanning; Krl=0, Krg>0
Figure 3-13: The impact of injection boundary model on defending phase connectivity and the implications for steady-state relative permeability calculation. Krl=liquid relative permeability and Krg=gas relative permeability.
Chapter 3 - Description of the Pore Network Simulator
110
a b c
d e f
Figure 3-14: Schematic representation of the dynamic model of full-face gas injection. rn denotes
real dangling pores and in the corresponding imaginary neighbors. (a) Onset of injection, each
dangling inlet (real) pore connected to an imaginary neighbors of equal size, (b) gas injected into
the imaginary pores (on a volume weighted bases) displaces water menisci in the real pores at a
rate determined by pore conductivity, (c) Three inlet pores completely gas-filled and hence
disconnected from their imaginary neighbours and gas now injected directly into the real pores, (d)
Four inlet pores now disconnected from their imaginary neighbours and injection into pore i1, the
imaginary neighbor of r1, halted since the equilibrium volume of the total gas injected into i1 equals
2 times the pore volume of i1 (or r1), (e) Gas injection and expansion continues, r1 still partially filled
thus maintaining network-wide water connectivity, Krw>0, (f) Krw>0 and gas expansion continues.
Chapter 3 - Description of the Pore Network Simulator
111
3.10 Modelling Spontaneous Gravity Migration
When buoyancy forces become sufficiently large so as to spontaneously overcome the
local capillary forces β e.g. in the post injection phase of CO2 sequestration, flow in
fractures, flow in reservoir regions far away from gravity-destabilized gas injection points,
or the evolution of solution gas during depressurization β a spontaneous mobilization or
migration of gas is triggered. The driving pressure for such a migration event can be
approximated by
βπππππ = βππβππ β
2ππππ π
ππ (3-26)
where, all terms are as previously defined.
The process of advancing gas-fluid interface positions follows the same steps as described
above (additional details can be found in Bondino et al, 2005; and Ezeuko et al, 2009). Gas
migration is now characterised by two displacement fronts however: the invasion front
(drainage) and the retraction front (imbibition). The imbibition process is assumed
spontaneous and modelled after the observations of Lenormand and Zarcone (1984,
1983). It is governed by local network water-gas topology, and hydrostatic and capillary
pressure considerations. We define a new relationship for imbibition, whereby the
pressure difference required to spontaneously imbibe water into a pore, π, filled with gas
is given by the relation:
βπππππ = π ππβ [1.0 + πΌπ] + βππ
β (3-27)
Here, βππβ is as defined in Equation (3-9), and πΌ is a topological parameter and is defined
as
πΆ= max[ πΆπ, πΆπ] (3-28)
where, πΌπ and πΌπ represent the fractional number of pore neighbours on either
sides (π and π) of pore π that are filled with water.
πΌπ = πππ€/ππ
π€+π
πΌπ = πππ€/ππ
π€+π
Chapter 3 - Description of the Pore Network Simulator
112
For a dangling gas pore (connected to its parent cluster via single connection), πΌ = 1, and
imbibition is piston-like. Snap-off type imbibition results when πΌ = 0, while for 0 < πΌ <
1, the imbibition is intermediate. Thus the greater the number of water-filled pores
neighbouring a gas pore, the more likely it is to be imbibed.
Imbibition mechanisms intermediate between piston and snap-off types have similar
features to the I1 and I2 imbibition types described by Lenormand and Zarcone (1984) as
shown in Figure 3-15.
Figure 3-15: The various mechanisms for meniscus displacement by imbibition (Lenormand and Zarcone, 1984). I1 means gas occupies only one pore, just like in the piston-type mechanism but the imbibition process takes longer in this case. That is, if the choice of an imbibition step were between a piston-type and an I1 type, the piston-type will occur first. I2 means gas occupies two adjacent pores. I2 is also a slow mechanism.
3.10.1 Mass Conservation during Buoyancy-Driven Migration
Because the liquid re-imbibition process during buoyancy-driven migration is modelled
discretely (there is no meniscus tracking as for drainage processes at the leading
migration front), care must be taken to ensure that mass is conserved during migration.
Figure 3-16 illustrates the schematic of how mass conservation is handled by the model
during migration.
Consider a gas cluster undergoing spontaneous migration in a water-saturated network.
The main stages of a migration step may be outlined as follows:
Stage 1
Chapter 3 - Description of the Pore Network Simulator
113
In stage 1, there are two perimeter water-filled pores with positive net entry pressures
located near the top of the migrating cluster (see Figure 3-16 with the aid of figure
legend).
Stage 2
In stage 2 the cluster invades the perimeter pore with the minimum filling time (p1 in
stage 1 of Figure 3-16), converting it into a gas-filled pore. The volume of the cluster has
now increased by an amount, πππ1, equal to the volume of the pore just invaded. An equal
volume of water must therefore re-imbibe the cluster to conserve its initial volume. Note
also that the new gas pore (pore π) is temporarily empty i.e. does not contain any mass of
gas at all (πππ₯ = 0). Before moving on to the re-imbibition stage (stages 3a to 3c in Figure
3-16) the total amount of oil that needs to be re-imbibed to conserve the volume of
cluster π (denoted as πππππ(π₯)
) must be updated.
Stage 3
In stage 3a, all the pores of cluster π, except pore π, are examined to check which has the
largest imbibition threshold as specified by Equation (3-27). In this example, gas-filled
pore ππ meets this criteria, but before it is re-imbibed a second check is made to see
whether its volume is less than the total amount of oil that needs to be re-imbibed to
conserve the volume of cluster π (i.e. πππππ(π₯)
> πππ1 ?). If so, pore ππ is spontaneously filled
with oil, its gas content (πππ1) is then transferred into pore π, and ππππ
π(π₯) is updated
accordingly (Figure 3-16).
It is often necessary to re-imbibe more than one gas pore during a migration step in order
to satisfy the volume conservation criteria since the smallest pores are more likely to be
re-imbibed first. Hence, the procedure in stage 3a is repeated in stage 3b, but a different
gas-filled pore, pore ππ, is re-imbibed this time. At stage 3c, the current value of πππππ(π₯)
is
less than πππ3 and this signals the end of the water re-imbibition for the present migration
step. The total mass of gas in cluster π is re-distributed in its constituent pores on a
volume weighted basis, whilst the current πππππ(π₯)
is carried over to the next migration step
(which returns to Stage 1 in Figure 3-16).
Chapter 3 - Description of the Pore Network Simulator
114
The average margin of error in mass conservation is of the order 0.0001%, depending on
the pore density of a migrating cluster β the larger the number of pores per migrating
cluster the more accurate the mass conservation.
Stage 1
Stage 2
Stage 3
(a) (b) (c)
Figure 3-16: Schematic of a buoyancy-driven migration process highlighting the steps for free gas conservation during liquid re-imbibition.
Chapter 3 - Description of the Pore Network Simulator
115
3.11 Gas Production Boundary Options
The expansion of gas inside a network initially saturated with water of negligible
compressibility implies the expulsion of a volume of water β via the network outlet β
equal to the amount of gas expansion. In the initial stages, only water is produced. But as
gas continues to expand and displace water, it eventually arrives at the outlet and must
be produced (unless a closed system boundary is being modelled). The flexibility afforded
in network models to locally control pore-level events allows us to consider different gas
production boundary options depending on the flow conditions. Two options for
producing gas upon breakthrough have been implemented in the model.
Buffer option: the last few rows of the water-filled pores at outlet end of the network are
designated as a production buffer. The invasion of any of the buffer pores signals gas
breakthrough and the gas is immediately withdrawn from the buffer pore just invaded
and the pore spontaneously re-imbibed with water. The instantaneous production rate is
the cumulative volume of gas withdrawn from the buffer over the timestep. Under
capillary dominated flow, gas will typically be produced through only one buffer pore. The
invasion of a buffer pore means that all other non-buffer perimeter pores in the network
had larger entry thresholds and the first buffer pore to be invaded is repeatedly re-
imbibed and invaded. Thus no further saturation change occurs inside the network after
breakthrough β an example of the classic capillary end effect. Under buoyancy-driven
migratory flow or in a viscous dominated flow at high rates, however, a buffer production
port leads to the artificial accumulation of gas beneath the buffer (Figure 3-17). These
boundary effects could be avoided by using larger networks and then restricting analysis
to regions of the network away from the buffer although this would be highly CPU
intensive.
Dynamic option: this option enables the use of smaller networks by mitigating production
boundary effects. Gas is produced through the dangling pores at the outlet of the
network. However, gas is not simply spontaneously removed once it reaches the outlet
pores but allowed to flow out at a rate that depends on the pore conductivity and the net
driving pressure across the pore using Poiseulleβs law. This obviates the need for a buffer.
Chapter 3 - Description of the Pore Network Simulator
116
For an outlet pore π connected to a network-spanning cluster π, the gas flow rate through
it is given as
ππ = ππ(ππππ
+ βππ£ππ π β ππ§) (3-29)
where, ππ is the pore conductivity, ππππ
the pressure of the parent cluster π of gas
pore π, βππ£ππ π the local viscous pressure drop across π, ππ§ the ambient system
pressure.
For an outlet gas pore π connected to a non-network spanning migrating cluster π, gas
production through it is given as
ππ = ππ(βππππππ
) (3-30)
where, βππππππ
is the net buoyancy pressure on pore π. In this case, a water re-
imbibition procedure (discussed shortly in detail with respect to a gas dissolution
process) is also called to re-adjust the volume of the gas cluster migrating out of
the network.
The total gas production over a timestep is simply the cumulative sum of productions
through all the outlet pores.
Figure 3-17 compares the two gas production options under different flow regimes. The
two options give identical results under a capillary dominated flow but considerably
different outcomes under a buoyancy-driven dispersive flow.
Chapter 3 - Description of the Pore Network Simulator
117
Buffer option Dynamic option
Stable regime
Migratory regime
Figure 3-17: The importance of gas production boundary option to the analysis of gas saturation patterns in stable and migratory regimes.
3.12 Gas Dissolution in Liquid
The dissolution of free gas into an undersaturated liquid during an injection process is
modelled in an analogous way as the liberation of dissolved gas from solution during
pressure depletion but with the direction of mass transfer across the gas-liquid interface
reversed. Gas bubbles act as gas (dissolved) sinks during liberation but as gas sources
during dissolution. The main steps in a dissolution process within a timestep are
summarized in Figure 3-18.
Consider the dissolution of CO2 into brine. First, we calculate the CO2 solubility in brine
given current temperature, pressure and brine salinity. The βtransferable gas massβ (TGM)
is then calculated as the mass of gas required to equilibrate the brine, given its current
concentration. Next, the gas concentration of the thin layer of water surrounding the gas-
water interface is set to the equilibrium concentration at current system pressure and
temperature, setting up a dissolved gas concentration gradient that then drives the
diffusion process over the course of the timestep. Mass transfer across the gas-water
interface during diffusion is shut off when TGM reaches zero or when the total amount of
gas contained in all the gas clusters is used up. Note that diffusion will continue in the
water phase for the remainder of the time and its concentration will therefore tend to
equilibrate across the system. The last step involves shrinking the depleted free gas to its
Chapter 3 - Description of the Pore Network Simulator
118
equilibrium volume (given its compressibility factor at the current system pressure and
temperature) by imbibing the requisite amount of liquid (absolute liquid pressure is
assumed to be constant). The liquid imbibition process is governed by pore capillary entry
thresholds, local gas-liquid interface topologies and hydrostatic pressure considerations
according to Equation (3-27).
1. Calculate gas solubility (Equivalent of equilibrium GOR in depressurization)
2. Calculate transferable gas mass to be dissolved
3. Transfer gas into liquid via Fickian Diffusion
4. Collapse gas bubbles to fit new PVT volume given mass left over after diffusion
Figure 3-18: The main algorithmic steps followed by the model during the dissolution of gas in liquid. The network schematics included in steps 3 and 4 show dissolved gas concentration distribution in the gaseous and liquid phases following diffusion and liquid re-imbibition processes, respectively. Blue delineates the boundary of the free gas phase which is assumed to enclose a region at the maximum (equilibrium) gas concentration, whilst lighter colours represent dissolved gas at varying concentrations.
Chapter 3 - Description of the Pore Network Simulator
119
3.13 Dissolved Gas Tracking during Buoyancy-Driven Migration Re-imbibed pores during migration must be assigned a dissolved gas concentration. The
simplest way to accomplish this is to assign to them the average dissolved gas
concentration in the network. However, because during migration free gas saturation
front can move rapidly ahead of the dissolved gas concentration front, assigning an
average network concentration to re-imbibed pores will systematically and irreversibly
alter the material balance in the system. Another option might be to assign to a re-
imbibed pore the concentration of its water-filled neighbours. But not all gas pores that
satisfy the imbibition criteria are guaranteed to have at least one water-filled neighbor.
The dissolved gas partitioning scheme implemented here involves assigning to the re-
imbibed pore the average concentration of the water-filled pores surrounding the gas
cluster just re-imbibed (this includes both perimeter and non-perimeter pores). It was
found to give a more consistent dissolved gas conservation.
3.14 CO2 Property Modelling
CO2 exhibits complex phase behaviour in response to changes in temperature and
pressure. These changes are highly non-linear and are difficult to extrapolate in a
straightforward manner. To facilitate predictive simulations of CO2 flow behaviour at
varying temperature and pressure conditions, known thermodynamic models of CO2
compressibility factor, CO2 density and CO2 solubility in brine are implemented in the
simulator.
3.14.1 A model of CO2 Density and Compressibility Factor
The CO2 compressibility factor is first calculated and then subsequently used to estimate
CO2 density. We start by calculating the molar volume of CO2, v, at a given temperature
and pressure using the Redlich-Kwong (1949) equation of state (Equation 3-31), which is a
modified form of the original van der Waal equation.
P= RT v-bβ - a T0.5v(v+b)β (3-31)
where, a = 7.54 Γ 107 β 4.13 Γ 104π[π] and it corrects for the intermolecular
interactions with changes in temperature; b = 27.80Β±0.01 [cm3mol-1] is the
Chapter 3 - Description of the Pore Network Simulator
120
effective volume of the molecules contained in a mole of gas; P=pressure [bar],
T=temperature [K], R = 83.144 [cm3βbarβKβ1molβ1] = universal gas constant
The parameters a and b were obtained by fitting experimental data (Spycher at al, 2003)
and are applicable in the temperature range 10oC β 107oC.
Recasting Equation (3-31) yields,
v3- RT Pβ v2-(RT Pβ - a T0.5Pβ +b2)v- ab T0.5Pβ =0 (3-32)
Equation (3-32) is solved by a cubic equation solver β solution yields one root if the
temperature and pressure condition is above the critical point of CO2 (1070.38psia,
31.1oC), but two or three roots if below the critical point. If there is more than one root,
the minimum root gives the molar volume of the liquid phase whereas the molar volume
of the gas phase (or the vapour phase) is given by the maximum root (Spycher at al,
2003).
The compressibility factor is then computed by the following equation;
Z= Pv RTβ (3-33)
Once Z is known then the density can be approximated according to
Ο = PM/ZRT (3-34)
where Ο is CO2 density [Kgm3], M = 0.044 [kg] = CO2 molar mass, and
R = 8.314472x10β5 [π3βbππβπΎβ1πππβ1]
Figure 3-19 shows how the Z computed with other methods compares against the
National Institute of Standards and Technology (NIST) data (Marini, 2007), whilst Figure
3-20 and Figure 3-21 show model implementation of CO2 compressibility factor (Z) and
CO2 density, respectively, as functions of pressure and temperature.
Chapter 3 - Description of the Pore Network Simulator
121
Figure 3-19: Compressibility factor of CO2 computed at 50oC and variable pressures using the van der Waal equation and the Redlch-Kwong equation parameterized in two ways (in the upper case the parameters π and π are constants; in the lower case parameter π is a constant whilst π is a function of temperature). The molar volumes recommended by NIST (data from Lemmon et al., 2003) are shown (squares). (Adapted from Marini, 2007)
Figure 3-20: Model implementation of CO2 compressibility factor (Z) as a function of pressure and
temperature from Redlich-Kwong (1949) equation of state.
10
40 50
70
90
110
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 100 200 300 400 500 600
CO
2 C
om
pre
ssib
ility
fac
tor
(Z)
Pressure, bar
Chapter 3 - Description of the Pore Network Simulator
122
Figure 3-21: Model implementation of CO2 density as a function of pressure and temperature
from Redlich-Kwong (1949) equation of state.
3.14.2 A model of CO2 Solubility in Brine
The simulator implements the model of Duan et al (2006), an empirical model for
calculating CO2 solubility in aqueous solutions containing Na+, K+, Ca2+, Mg2+, Cl-, and
SO42. The wide range of temperature, pressure and salinity conditions (273 to 533 K, 0 to
2000 bar, and 0 to 4.5 molality of salts) covered by the Duan et al model explains its
popularity in the CO2 storage modelling community, even aside from its other advantages
which include the following:
1. Its equations are non-iterative which makes the model straightforward to adopt
for reactive flow modelling
2. It easily handles CO2 solubility in brine containing multiple ions, namely: Na+, K+,
Ca2+, Mg2+, Cl-, and SO42-
3. The model is impressively accurate because of the extensive amount of
experimental data used in its parameterization. It is not only able to reproduces all
the reliable data used in its parameterization but has also predicted experimental
data not included in its parameterization.
From Duan and Sun (2003), the solubility of CO2 in brine can be calculated using
0
200
400
600
800
1000
1200
10 30 50 70 90 110
De
nsi
ty,
Kg/
m3
Temperature, oC
Gas
Liquid
Chapter 3 - Description of the Pore Network Simulator
123
lnmCO2 = lnyCO2
ΟCO2P - ΞΌ1(0)
CO2RT β - 2Ξ»CO2-Na(mNa+mK+2mCa+2mMg)
- ΞΆCO2-Na-ClmCl(mNa+mK+mCa+mMg) + 0.07mSO4 (3-35)
where, mCO2 is the CO2 solubility in brine [mol/kg], T the absolute temperature
[K], P the total pressure of the system [bar], R the universal gas constant, mi the
molality of components i dissolved in water, π¦πΆπ2 the mole fraction of CO2 in the
vapour phase, ππΆπ2 the fugacity coefficient of CO2, π1(0)
πΆπ2 the standard
chemical potential of CO2 in liquid phase, ππΆπ2βNa the interaction parameter
between CO2 and Na+, ππΆπ2βNaβCl the interaction parameter between CO2 and
Na+, Cl-
All parameters presented in Equation (3-35) can be calculated directly without any
iteration as follows:
1. Mole fraction of CO2 in vapor phase
yCO2=(P - PH2O)/P (3-36)
where P is the pressure of the system and ππ»2π is the pure water pressure, which
can be taken from the steam tables (Haar et al., 1984) or can be calculated from
an empirical equation given below.
P=(PcT/Tc) [1+c1(-t)1.9
+c2t+c3t2+c4t3+c5t4] (3-37)
where π is temperature [K], π‘ = (π β ππ)/ππ, ππ and ππ the critical temperature
and critical pressure of water, respectively (ππ= 647.29 K, ππ= 220.85 bar).
The parameters of Equation (3-37), c1βc5, are listed in Table 3-1.
Chapter 3 - Description of the Pore Network Simulator
124
Table 3-1: Parameters of Equation (3-37)
C1 -38.640844 C2 5.8948420 C3 59.876516 C4 26.654627 C5 10.637097
2. The interaction parameters: ion interaction parameters (ππΆπ2βNa and ππΆπ2βNaβCl)
and the dimensionless standard chemical potential (ππΆπ21(0) π πβ ).
Following Pitzer et al. (1984), the following equation was used for the interaction
parameters as functions of temperature and pressure,
Par(T,P) = c1+c2T+ c3 Tβ +c4T2+ c5 (630-T)β +c6P+c7PlnT+ c8P Tβ + c9P (630-T)β
+ c10P2 (630-T)2
β +c11TlnP (3-38)
The parameters c1, c2, c3, etc β¦. In Equation (3-38) are listed in 3-2.
Table 3-2: Parameters of Equation (3-38)
T-P Coefficient ππΆπ21(0) π πβ ππΆπ2βNa ππΆπ2βNaβCl
C1 28.9447706 -0.411370585 3.36389723e-4 C2 -0.0354581768 6.07632013e-4 -1.98298980e-5 C3 -4770.67077 97.5347708 C4 1.02782768e-5 C5 33.8126098 C6 9.04037140e-3 C7 -1.14934031e-3 C8 -0.307405726 -0.0237622469 2.12220830e-3 C9 -0.0907301486 0.0170656236 -5.24873303e-3 C10 9.32713393e-4 C11 1.41335834e-5
3. CO2 fugacity coefficient
ΟCO2=c1+[c2+c3T+ c4 Tβ + c5 (T-150)β ]P+[c6+c7T+ c8 Tβ ]P2 + [c9 + c10T +
c11 Tβ ]lnP + [c12 + c13T] Pβ + c14 Tβ + c16T2 (3-39)
where, T is the temperature [K] and P the pressure [bar].
Chapter 3 - Description of the Pore Network Simulator
125
The parameters c1, c2, c3, etc, in Equation (3-39) were fitted to ππΆπ2calculated from the
EOS of Duan et al. (1992) at the TβP range where this CO2 solubility model is valid. Table
3-3 lists the parameters for Equation (3-39).
Table 3-3: Parameters of equation 11 Par T-P range
1 2 3 4 5 6
C1 1.0 -7.1734882E-1 -6.5129019E-2 5.0383896 -16.063152 -1.5693490E-1 C2 4.7586835E-3 1.5985379E-4 -2.1429977E-4 -4.4257744E-3 -2.7057990E-3 4.4621407E-4 C3 -3.3569963E-6 -4.9286471E-7 -1.1444930E-6 0.0 0.0 -9.1080591E-7 C4 0.0 0.0 0.0 1.9572733 1.4119239E-1 0.0 C5 -1.3179396 0.0 0.0 0.0 0.0 0.0 C6 -3.8389101E-6 -2.7855285E-7 -1.1558081E-7 2.4223436E-6 8.1132965E-7 1.0647399E-7 C7 0.0 1.1877015E-9 1.1952370E-9 0.0 0.0 2.4273357E-10 C8 2.2815104E-3 0.0 0.0 -9.3796135E-4 -1.1453082E-4 0.0 C9 0.0 0.0 0.0 -1.5026030 2.3895671 3.5874255E-1 C10 0.0 0.0 0.0 3.0272240E-3 5.0527457E-4 6.3319710E-5 C11 0.0 0.0 0.0 -31.377342 -17.763460 -249.89661 C12 -96.539512 -221.34306 -12.847063 985.92232 0.0 C13 4.4774938E-1 0.0 0.0 0.0 0.0 C14 101.81078 71.820393 0.0 0.0 888.76800 C15 5.3783879E-6 6.6089246E-6 -1.5056648E-5 -5.4965256E-7 -6.6348003E-7
1: 273K < T < 573K, P < Pl (when T < 305K, Pl equals to the saturation pressure of CO2; when 305K < T < 405K, Pl =75 + (T - 305)Γ1.25; when T > 405K, Pl =200bar.); 2: 273K < T < 340K, Pl < P < 1000bar; 3: 273K < T < 340K, P > 1000bar; 4: 340K < T < 435K, Pl < P < 1000bar; 5: 340K < T < 435K, P > 1000bar; and 6: T > 435K, P > Pl. Par=parameters.
4. Calculate CO2 Solubility in brine
Finally, for a given temperature, pressure and brine salinity (given by the molalities of the
ions Na+, K+, Ca2+, Mg2+, Cl-, and SO42- ) the solubility of CO2 in brine is calculated using
Equation (3-35) in mols/kg and subsequently converted to kg/m3 (as Kg of CO2 per m3 of
brine).
Figure 3-22 shows a comparison of CO2 solubility in brine (1m NaCl) as a function of
pressure at 50oC and 100oC as calculated by the current simulator (PNM), and an output
from an implementation of the same model by Duan et al (2006). The agreement
between the two implementations is excellent. Figure 3-23 shows model implementation
of CO2 solubility in salt-free brine as a function of pressure and temperature.
Chapter 3 - Description of the Pore Network Simulator
126
Figure 3-22: CO2 Solubility in 1m NaCl brine @ 50oC and 100
oC
Figure 3-23: Model implementation of CO2 solubility in salt-free brine as a function of pressure and temperature, using the model of Dual et al (2006).
3.15 Chapter Summary This chapter has described the constitutive physical principles and the implementation
algorithms that went into the development of a pore network model capable of
simulating the full range of gravity-driven and viscous-driven multi-phase flow in porous
media. The model extends the conceptual framework originally intended for evaluating
an internal drive process (solution gas drive) to an external drive (injection) process. This
involved the modification of several simulator modules and the development of new
algorithms that dealt largely with the implementation of process physics relevant to
100oC
50oC
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0 100 200 300 400 500 600
CO
2 S
olu
bilt
y (m
ol/
kg)
Pressure (bar)
CO2 Solubility in 1m NaCl brine
PNM
Duan et al (2006)
0
20
40
60
80
100
10 30 50 70 90 110
Solu
bili
ty,
[Kg(
CO
2)/
m3
(H2
O)]
o
r [K
g(C
O2
)/1
00
0K
g(H
2O
) , i
f d
en
sity
of
wat
er
~10
00
kg/m
3]
Temperature, oC
Pressure(MPa)
Critical point
Supercritical boundary
Chapter 3 - Description of the Pore Network Simulator
127
external drive processes β such as the flow of multiple phases across the inlet face of the
network, the coupling of dynamic viscous pressure gradients to capillary and gravity
forces, and the isothermal dissolution of an injected phase. Modifications that implement
enhanced drainage and imbibition criteria, as well as new production boundary options
during stable and unstable flow were also presented. Other model modifications include
improved procedures for the conservation of free and dissolved gas mass during
buoyancy-driven spontaneous migration. Finally, some relevant models for predicting CO2
compressibility factor, density and solubility that furnish the simulator with the flexibility
to investigate CO2 flow behavior in brine or oil over a wide range of temperature,
pressure and salinity conditions were presented.